共查询到20条相似文献,搜索用时 58 毫秒
1.
M. C. Zdun 《Aequationes Mathematicae》2001,61(3):239-254
Summary. We investigate the bounded solutions j:[0,1]? X \varphi:[0,1]\to X of the system of functional equations¶¶j(fk(x))=Fk(j(x)), k=0,?,n-1,x ? [0,1] \varphi(f_k(x))=F_k(\varphi(x)),\;\;k=0,\ldots,n-1,x\in[0,1] ,(*)¶where X is a complete metric space, f0,?,fn-1:[0,1]?[0,1] f_0,\ldots,f_{n-1}:[0,1]\to[0,1] and F0,...,Fn-1:X? X F_0,...,F_{n-1}:X\to X are continuous functions fulfilling the boundary conditions f0(0) = 0, fn-1(1) = 1, fk+1(0) = fk(1), F0(a) = a,Fn-1(b) = b,Fk+1(a) = Fk(b), k = 0,?,n-2 f_{0}(0) = 0, f_{n-1}(1) = 1, f_{k+1}(0) = f_{k}(1), F_{0}(a) = a,F_{n-1}(b) = b,F_{k+1}(a) = F_{k}(b),\,k = 0,\ldots,n-2 , for some a,b ? X a,b\in X . We give assumptions on the functions fk and Fk which imply the existence, uniqueness and continuity of bounded solutions of the system (*). In the case X = \Bbb C X= \Bbb C we consider some particular systems (*) of which the solutions determine some peculiar curves generating some fractals. If X is a closed interval we give a collection of conditions which imply respectively the existence of homeomorphic solutions, singular solutions and a.e. nondifferentiable solutions of (*). 相似文献
2.
On the iterates of Euler's function 总被引:1,自引:0,他引:1
R. Warlimont 《Archiv der Mathematik》2001,76(5):345-349
Asymptotic representations are given for the three sums ?n £ x j(n)/j(j(n))\textstyle\sum\limits \limits _{n\le x} \varphi (n)/\varphi \bigl (\varphi (n)\bigr ), ?n £ x j(n)/j(j(n))\textstyle\sum\limits \limits _{n\le x} \varphi (n)/\varphi \bigl (\varphi (n)\bigr ), ?n £ x log j(n)/j(j(n)) ; j\textstyle\sum\limits \limits _{n\le x}\ \log \, \varphi (n)/\varphi \bigl (\varphi (n)\bigr )\ ; \ \varphi is Euler's function. 相似文献
3.
Let Δ be a simplicial complex on V = {x 1, . . . , x n }, with Stanley–Reisner ideal ${I_{\Delta}\subseteq R=k[x_1,\ldots, x_n]}Let Δ be a simplicial complex on V = {x
1, . . . , x
n
}, with Stanley–Reisner ideal ID í R=k[x1,?, xn]{I_{\Delta}\subseteq R=k[x_1,\ldots, x_n]} . The goal of this paper is to investigate the class of artinian algebras A=A(D,a1,?,an) = R/(ID,x1a1,?,xnan){A=A(\Delta,a_1,\ldots,a_n)= R/(I_{\Delta},x_1^{a_1},\ldots,x_n^{a_n})} , where each a
i
≥ 2. By utilizing the technique of Macaulay’s inverse systems, we can explicitly describe the socle of A in terms of Δ. As a consequence, we determine the simplicial complexes, that we will call levelable, for which there exists a tuple (a
1, . . . , a
n
) such that A(Δ, a
1, . . . , a
n
) is a level algebra. 相似文献
4.
Constantin Costara 《Integral Equations and Operator Theory》2012,73(1):7-16
Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let rT(x) = limsupn ? ¥ || Tnx|| 1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r
T
(x) = 0 if and only if rj(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r
T
(By) = 0 if and only if rj( T) (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}. 相似文献
5.
G. I. Perel'muter 《Mathematical Notes》1975,18(3):840-844
Supposef(x1,..., xn) is a polynomial of even degree d having coefficients in the finite field k=[q] and satisfying certain natural conditions, and let χ be the quadratic character of k. Then $$\left| {\sum {x_1 , \ldots ,} x_n \in k\chi (f(x_1 , \ldots ,x_n ))} \right| \leqslant Cq^{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} $$ where the constant C depends only on d and n. 相似文献
6.
Demetris Hadjiloucas 《Monatshefte für Mathematik》2009,50(5):151-178
In this article, we extend the recently developed abstract theory of universal series to include averaged sums of the form
\frac1f(n)?j=0n aj xj{\frac{1}{\phi(n)}\sum_{j=0}^{n} a_j x_j} for a given fixed sequence of vectors (x
j
) in a topological vector space X over a field
\mathbbK{\mathbb{K}} of real or complex scalars, where (f(n)){(\phi(n))} is a sequence of non-zero field scalars. We give necessary and sufficient conditions for the existence of a sequence of coefficients
(a
j
) which make the above sequence of averaged sums dense in X. When applied, the extended theory gives new analogues to well known classical theorems including those of Seleznev, Fekete
and Menchoff. 相似文献
7.
For n = 1, the space of ${\mathbb{R}}For n = 1, the space of
\mathbbR{\mathbb{R}} -places of the rational function field
\mathbbR(x1,?, xn){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of
\mathbbR{\mathbb{R}} -places of the rational function field
\mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of
\mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of
\mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of
\mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable. 相似文献
8.
For x = (x
1, x
2, …, x
n
) ∈ (0, 1 ]
n
and r ∈ { 1, 2, … , n}, a symmetric function F
n
(x, r) is defined by the relation
Fn( x,r ) = Fn( x1,x2, ?, xn;r ) = ?1 \leqslant1 < i2 ?ir \leqslant n ?j = 1r \frac1 - xijxij , {F_n}\left( {x,r} \right) = {F_n}\left( {{x_1},{x_2}, \ldots, {x_n};r} \right) = \sum\limits_{1{ \leqslant_1} < {i_2} \ldots {i_r} \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 - {x_{{i_j}}}}}{{{x_{{i_j}}}}}} }, 相似文献
9.
Jan A. van Casteren 《Journal of Evolution Equations》2011,11(2):457-476
Let
(tj)j ? \mathbbN{\left(\tau_j\right)_{j\in\mathbb{N}}} be a sequence of strictly positive real numbers, and let A be the generator of a bounded analytic semigroup in a Banach space X. Put
An=?j=1n(I+\frac12 tjA) (I-\frac12 tjA)-1{A_n=\prod_{j=1}^n\left(I+\frac{1}{2} \tau_jA\right) \left(I-\frac{1}{2} \tau_jA\right)^{-1}}, and let x ? X{x\in X}. Define the sequence
(xn)n ? \mathbbN ì X{\left(x_n\right)_{n\in\mathbb{N}}\subset X} by the Crank–Nicolson scheme: x
n
= A
n
x. In this paper, it is proved that the Crank–Nicolson scheme is stable in the sense that
supn ? \mathbbN||Anx|| < ¥{\sup_{n\in\mathbb{N}}\left\Vert A_nx\right\Vert<\infty}. Some convergence results are also given. 相似文献
10.
Bernhard Burgstaller 《Monatshefte für Mathematik》2009,265(4):1-11
There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x
1, x
2, ... we have
limn ? ¥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra O2{\mathcal {O}_2} such that for an independent sequence x
1, x
2, ... of μ-distributed random elements x
i
we have
lim infn ? ¥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1]
d
) is d. 相似文献
11.
DingHua Yang 《中国科学A辑(英文版)》2009,52(10):2287-2308
Using the axiomatic method,abstract concepts such as abstract mean, abstract convex function and abstract majorization are proposed. They are the generalizations of concepts of mean, convex function and majorization, respectively. Through the logical deduction, the fundamental theorems about abstract majorization inequalities are established as follows: for arbitrary abstract mean Σ and Σ , and abstract Σ → Σ strict convex function f(x) on the interval I, if xi, yi ∈ I (i = 1, 2, . . . , n) satisfy that (x1... 相似文献
12.
13.
L. Losonczi 《Aequationes Mathematicae》1999,58(3):223-241
Summary. Let F, Y \Phi, \Psi be strictly monotonic continuous functions, F,G be positive functions on an interval I and let n ? \Bbb N \{1} n \in {\Bbb N} \setminus \{1\} . The functional equation¶¶F-1 ([(?i=1nF(xi)F(xi))/(?i=1n F(xi)]) Y-1 ([(?i=1nY(xi)G(xi))/(?i=1n G(xi))]) (x1,?,xn ? I) \Phi^{-1}\,\left({\sum\limits_{i=1}^{n}\Phi(x_{i})F(x_{i})\over\sum\limits_{i=1}^{n} F(x_{i}}\right) \Psi^{-1}\,\left({\sum\limits_{i=1}^{n}\Psi(x_{i})G(x_{i})\over\sum\limits_{i=1}^{n} G(x_{i})}\right)\,\,(x_{1},\ldots,x_{n} \in I) ¶was solved by Bajraktarevi' [3] for a fixed n 3 3 n\ge 3 . Assuming that the functions involved are twice differentiable he proved that the above functional equation holds if and only if¶¶Y(x) = [(aF(x) + b)/(cF(x) + d)], G(x) = kF(x)(cF(x) + d) \Psi(x) = {a\Phi(x)\,+\,b\over c\Phi(x)\,+\,d},\qquad G(x) = kF(x)(c\Phi(x) + d) ¶where a,b,c,d,k are arbitrary constants with k(c2+d2)(ad-bc) 1 0 k(c^2+d^2)(ad-bc)\ne 0 . Supposing the functional equation for all n = 2,3,... n = 2,3,\dots Aczél and Daróczy [2] obtained the same result without differentiability conditions.¶The case of fixed n = 2 is, as in many similar problems, much more difficult and allows considerably more solutions. Here we assume only that the same functional equation is satisfied for n = 2 and solve it under the supposition that the functions involved are six times differentiable. Our main tool is the deduction of a sixth order differential equation for the function j = F°Y-1 \varphi = \Phi\circ\Psi^{-1} . We get 32 new families of solutions. 相似文献
14.
R. D. Blyth 《Archiv der Mathematik》2002,78(5):337-344
Let n be an integer greater than 1, and let G be a group. A subset {x1, x2, ..., xn} of n elements of G is said to be rewritable if there are distinct permutations p \pi and s \sigma of {1, 2, ..., n} such that¶¶xp(1)xp(2) ?xp(n) = xs(1)xs(2) ?xs(n). x_{\pi(1)}x_{\pi(2)} \ldots x_{\pi(n)} = x_{\sigma(1)}x_{\sigma(2)} \ldots x_{\sigma(n)}. ¶¶A group is said to have the rewriting property Qn if every subset of n elements of the group is rewritable. In this paper we prove that a finite group of odd order has the property Q3 if and only if its derived subgroup has order not exceeding 5. 相似文献
15.
Let K be a convex body in
\mathbbRn \mathbb{R}^n with volume |K| = 1 |K| = 1 . We choose N 3 n+1 N \geq n+1 points x1,?, xN x_1,\ldots, x_N independently and uniformly from K, and write C(x1,?, xN) C(x_1,\ldots, x_N) for their convex hull. Let
f : \mathbbR+ ? \mathbbR+ f : \mathbb{R^+} \rightarrow \mathbb{R^+} be a continuous strictly increasing function and 0 £ i £ n-1 0 \leq i \leq n-1 . Then, the quantity¶¶E (K, N, f °Wi) = òK ?òK f[Wi(C(x1, ?, xN))]dxN ?dx1 E (K, N, f \circ W_{i}) = \int\limits_{K} \ldots \int\limits_{K} f[W_{i}(C(x_1, \ldots, x_N))]dx_{N} \ldots dx_1 ¶¶is minimal if K is a ball (Wi is the i-th quermassintegral of a compact convex set). If f is convex and strictly increasing and 1 £ i £ n-1 1 \leq i \leq n-1 , then the ball is the only extremal body. These two facts generalize a result of H. Groemer on moments of the volume of C(x1,?, xN) C(x_1,\ldots, x_N) . 相似文献
16.
R. Ger 《Aequationes Mathematicae》2000,60(3):268-282
Summary. Quite recently C. Alsina, P. Cruells and M. S. Tomás [2], motivated by F. Suzuki's property of isosceles trapezoids, have proposed the following orthogonality relation in a real normed linear space (X, ||·||) (X, \Vert \cdot \Vert) : two vectors x,y ? X x,y \in X are T-orthogonal whenever¶||z-x ||2 + ||z-y ||2 = ||z ||2 + ||z-(x+y) ||2 \Vert z-x \Vert^2 + \Vert z-y \Vert^2 = \Vert z \Vert^2 + \Vert z-(x+y) \Vert^2 ¶for every z ? X z \in X . A natural question arises whether an analogue of T-orthogonality may be defined in any real linear space (without a norm structure). Our proposal reads as follows. Given a functional j \varphi on a real linear space X we say that two vectors x,y ? X x,y \in X are j \varphi -orthogonal (and write x^jy x\perp_{\varphi}y ) provided that Dx,yj = 0 \Delta_{x,y}\varphi = 0 (Dh1,h2 \Delta_{h_1,h_2} stands here and in the sequel for the superposition Dh1 °Dh2 \Delta_{h_1} \circ \Delta_{h_2} of the usual difference operators).¶We are looking for necessary and/or sufficient conditions upon the functional j \varphi to generate a j \varphi -orthogonality such that the pair X,^j X,\perp_{\varphi} forms an orthogonality space in the sense of J. Rätz (cf. [6]). Two new characterizations of inner product spaces as well as a generalization of some results obtained in [2] are presented. 相似文献
17.
M. A. Taylor 《Aequationes Mathematicae》2000,60(3):283-290
Summary. It is shown that provided F and G are injective in every argument, the functional equation of generalized m ×n m \times n bisymmetry (m,n 3 2) (m,n \ge 2) ,¶¶ G(F1(x11, \hdots , x1n),\hdots , Fm(xm1,\hdots, xmn)) G(F_1(x_{11}, \hdots , x_{1n}),\hdots , F_m(x_{m1},\hdots, x_{mn})) ¶ = F(G1(x11,\hdots , xm1),\hdots , Gn(x1n,\hdots , xmn)) = F(G_1(x_{11},\hdots , x_{m1}),\hdots , G_n(x_{1n},\hdots , x_{mn})) ¶may be reduced to ¶¶ G([`(F)]1(u11, \hdots , u1n),\hdots ,[`(F)]m(um1,\hdots, umn)) G(\overline{F}_1(u_{11}, \hdots , u_{1n}),\hdots , \overline{F}_m(u_{m1},\hdots, u_{mn})) ¶ = F([`(G)]1(u11,\hdots , um1),\hdots ,[`(G)]n(u1n,\hdots , umn)) = F(\overline{G}_1(u_{11},\hdots , u_{m1}),\hdots ,\overline{G}_n(u_{1n},\hdots , u_{mn})) ¶where¶¶ Fi(xi1,\hdots , xin) = [`(F)]i (ji1(xi1),\hdots , jin(xin)), Gj(x1j, \hdots , xmj) = [`(G)]j(j1j (x1j),\hdots, jmj(xmj)) F_i(x_{i1},\hdots , x_{in}) = \overline{F}_i (\varphi_{i1}(x_{i1}),\hdots , \varphi_{in}(x_{in})), G_j(x_{1j}, \hdots , x_{mj}) = \overline{G}_j(\varphi_{1j} (x_{1j}),\hdots, \varphi_{mj}(x_{mj})) ,¶¶jij < /FORMULA > are surjections and < FORMULA > \varphi_{ij} are surjections and
18.
Motivated by certain questions in physics, Atiyah defined a determinant function which to any set of n distinct points x 1, . . . , x n in ${\mathbb R^3}$ assigns a complex number D(x 1, . . . , x n ). In a joint work, he and Sutcliffe stated three intriguing conjectures about this determinant. They provided compelling numerical evidence for the conjectures and an interesting physical interpretation of the determinant. The first conjecture asserts that the determinant never vanishes, the second states that its absolute value is at least one, and the third says that ${|D(x_1,\ldots, x_n)|^{n-2} \geq \prod_{i=1}^n |D(x_1,\ldots, x_{i-1},x_{i+1},\ldots, x_n)|}$ . Despite their simple formulation, these conjectures appear to be notoriously difficult. Let D n denote the Atiyah determinant evaluated at the vertices of a regular n-gon. We prove that ${\lim_{n\to \infty}\frac{\ln D_n}{n^2}=\frac{7\zeta(3)}{2\pi^2}-\frac{\ln 2}{2}=0.07970479\ldots}$ and establish the second conjecture in this case. Furthermore, we prove the second conjecture for vertices of a convex quadrilateral and the third conjecture for vertices of an inscribed quadrilateral. 相似文献
19.
Examples of factorial threefolds complete intersection in $${\mathbb{P}^5}$$ with many singularities
Pietro Sabatino 《Ricerche di matematica》2009,58(2):285-297
Denote by ν
m
(d) the maximal integer for which there exists for d >> 0{d \gg 0} a threefold
X ì \mathbbP5{X\subset \mathbb{P}^5} complete intersection of hypersurfaces of degree respectively d and d − 1 such that X has only ordinary singularities of order m and |Sing(X)| = ν
m
(d). We prove that, nm(d) 3 j(d){\nu_m(d)\ge \varphi(d)} where j(d) ~ d5{\varphi(d)\sim d^5} asymptotically. This result extends (Di Gennaro and Franco in Commun Contemp Math 10(5):745–764, 2008, Corollary 2.10). 相似文献
20.
The Essential Norm of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions
Olli Hyv?rinen Mika Kemppainen Mikael Lindstr?m Antti Rautio Erno Saukko 《Integral Equations and Operator Theory》2012,72(2):151-157
For a wide class of radial weights we calculate the essential norm of a weighted composition operator uCj{uC_\varphi} on the weighted Banach spaces of analytic functions in terms of the analytic function
u \colon \mathbb D ? \mathbb C{u \colon \mathbb D \to \mathbb C} and the nth power of the analytic selfmap j{\varphi} of the open unit disc
\mathbb D{\mathbb D} . We also apply our result to calculate the essential norm of composition operators acting on Bloch type spaces with general
radial weights. 相似文献
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